Structure of globally hyperbolic spacetimes-with-timelike-boundary

Abstract

Globally hyperbolic spacetimes-with-timelike-boundary $(\overline{M}=M\cup \partial M,g)$ are the natural class of spacetimes where regular boundary conditions (eventually asymptotic, if $\partial M$ is obtained by means of a conformal embedding) can be posed. $\partial M$ represents the naked singularities and can be identified with a part of the intrinsic causal boundary. Apart from general properties of $\partial M$, the splitting of any globally hyperbolic $(\overline{M},g)$ as an orthogonal product $\mathbb{R}\times \bar{\Sigma }$ with Cauchy slices-with-boundary $\{t\}\times \bar{\Sigma }$ is proved. This is obtained by constructing a Cauchy temporal function $\tau$ with gradient $\nabla \tau$ tangent to $\partial M$ on the boundary. To construct such a $\tau$, results on stability of both global hyperbolicity and Cauchy temporal functions are obtained. Apart from having their own interest, these results allow us to circumvent technical difficulties introduced by $\partial M$. The techniques also show that $\overline{M}$ is isometric to the closure of some open subset in a globally hyperbolic spacetime (without boundary). As a trivial consequence, the interior $M$ both splits orthogonally and can be embedded isometrically in some ${\mathbb{L}}^N$, extending so properties of globally spacetimes without boundary to a class of causally continuous ones.